The imaginary number "i" is defined to be the square root of -1. It is well known that the concept of the imaginary number "i" is essential in performing numerous mathematical, scientific and engineering calculations. When the value "i" is squared, the result is the negative integer one (-1); the value of "i" raised to the third power results in -i; and the value of "i" raised to the fourth power exponential results in the positive integer 1 (+1). Thereafter, the sequential values of "i" raised to successive continuous exponential powers repeat--e.g., "i" taken to the 5th exponential power equals "i"; "i" taken to the 6th exponential power equals negative one (-1); "i" taken to the 7th exponential power equals negative "-i" (-i); and "i" taken to the 8th exponential power equals the positive integer one (+1).
Equations derived from the repeating sequence of values of "i" taken to successive, continuous exponential powers, where "N" is a natural number, are: i.sup.4N =1; i.sup.4N+1 =i; i.sup.4N+2 =-1; and i.sup.4N+3 =-i. For example, based upon the above equations, i.sup.16 =i.sup.4(4) =1; i.sup.17 =i.sup.4(4)+1 =i; i.sup.18 =i.sup.4(4)+2 =-1; and i.sup.19 =i.sup.4(4)+3 =-i.
The above information and calculations concerning the repetitive values of the imaginary number "i" raised to successive continuous exponential powers is well known to the art. The following Matrix A has been formulated using the exponents cross-references with the four powers of "i".
______________________________________ MATRIX A ______________________________________ i 1 5 9 13 17 -1 2 6 10 14 18 -i 3 7 11 15 19 1 4 8 12 16 20 ______________________________________
The following Matrix B employs only digits that the numbers one to twenty have in common. It is noted that the number ten and twenty and every sequence of ten rotates in the -1 and 1 positions.
______________________________________ MATRIX B ______________________________________ i 1 5 9 3 7 -i 2 6 0 4 8 -i 3 7 1 5 9 1 4 8 2 6 0 ______________________________________
Matrix C illustrates only the first nine digits of Matrix B because the numbers ten, twenty, thirty, forty . . . influence the next rotating nine numbers.
______________________________________ MATRIX C ______________________________________ i 1 5 9 -1 2 6 -i 3 7 1 4 8 ______________________________________
The following Matrix D is expanded to include the influence of the numbers ten and twenty. The number ten includes the odd number one in the tens digit and ten is represented by -1 in Matrix A. The number twenty has the even number (i.e., 2) in the tens digit position, and the number twenty is represented by numeral 1 in Matrix A.
______________________________________ MATRIX D TENS DIGIT ONES DIGIT ______________________________________ i 1 5 9 odd = -1 -1 2 6 even = 1 -i 3 7 1 4 8 ______________________________________
The final version of the table must include the number zero. Referring to Matrix D, the number zero as a tens digit is even and is represented by the numeral 1. The number zero as a ones digit is positive one, which takes into consideration zero as the exponent in i.sup.0 =1 because any number raised to a 0 exponential power is defined as being 1. Matrix E, represented below, condenses the information derived from the preceding matrices.
______________________________________ MATRIX E TENS DIGIT ONES DIGIT ______________________________________ i 1 5 9 odd = -1 -1 2 6 even = 1 -i 3 7 1 0 4 8 ______________________________________
Examples of calculations of the imaginary number "i" raised to the different exponential powers based upon Matrix E are illustrated below.
______________________________________ TENS DIGIT ONES DIGIT MULTIPLY ANS. ______________________________________ i.sup.16 1 is odd = -1 6 = -1 (-1)(-1) 1 i.sup.17 1 is odd = -1 7 = -i (-1)(-i) i i.sup.18 1 is odd = -1 8 = 1 (-1)(1) -1 i.sup.19 1 is odd = -1 9 = i (-1)(i) -i i.sup.20 2 is even = 1 0 = 1 (1)(1) 1 ______________________________________
It is apparent from the above examples that Matrix E provides an alternative to the step of factoring the exponential powers to which the imaginary number "i" is raised. Matrix E advantageously is easy to remember and uses only the values of the tens and ones digit to solve the problem regardless of the magnitude of the exponent to which "i" is raised.
Although Matrix E eliminates the operation of factoring the exponential powers of "i", it is advantageous to eliminate any type of mental steps in the problem solution process. Matrix E discloses two variations for each power of "i", respectively for the ones digit indicating odd or even. If "d" represents the "odd" condition, and if "e" represents the "even" condition, the following Matrix F is derived.
______________________________________ MATRIX F ______________________________________ i e1 d3 e5 d7 e9 -1 d0 e2 d4 e6 d8 -i d1 e3 d5 e7 d9 1 e0 d2 e4 d6 e8 ______________________________________
A physical structure can be constructed to represent two variations without any type of mental process being employed by the user during the solution of a problem.
It is a primary object of the present invention to provide methods and apparatus for determining the value of the imaginary number "i" raised to any exponential power based upon the values of the ones and tens digit of the exponent to which the imaginary number "i" is being raised. The methods and apparatus of the present invention eliminate the step of factoring exponential powers, and provide the correct answer to the problem based exclusively on the value of the tens and ones digit of the exponent. Other advantages and features of the present invention will become apparent from the following further description of the invention in conjunction with the drawings.